Randomized neural networks for learning stochastic dependences

We consider the problem of learning the dependence of one random variable on another, from a finite string of independently identically distributed (i.i.d.) copies of the pair. The problem is first converted to that of learning a function of the latter random variable and an independent random variable uniformly distributed on the unit interval. However, this cannot be achieved using the usual function learning techniques because the samples of the uniformly distributed random variables are not available. We propose a novel loss function, the minimizer of which results in an approximation to the needed function. Through successive approximation results (suggested by the proposed loss function), a suitable class of functions represented by combination feedforward neural networks is selected as the class to learn from. These results are also extended for countable as well as continuous state-space Markov chains. The effectiveness of the proposed method is indicated through simulation studies.